# Even Number: An integer n is said to be even if it has the form n = 2k for some integer k. That is, n is even if and only if n divisible by 2.

Save this PDF as:

Size: px
Start display at page:

Download "Even Number: An integer n is said to be even if it has the form n = 2k for some integer k. That is, n is even if and only if n divisible by 2."

## Transcription

1 MATH 337 Proofs Dr. Neal, WKU This entire course requires you to write proper mathematical proofs. All proofs should be written elegantly in a formal mathematical style. Complete sentences of explanation are required. Do not simply write an equation; you must explain what the equation is giving and/or why it is being used. Moreover, all equations must be properly aligned with no scratch outs. Always give a conclusion. We will begin by reviewing several standard methods of proof using the following basic definitions and using the facts that the sum and product of integers are integers. Divisibility: We say that a divides b if there exists an integer k such that b = k a. For example, 6 divides 18 because 18 = 3! 6, where k = 3 is an integer. Even Number: An integer n is said to be even if it has the form n = 2k for some integer k. That is, n is even if and only if n divisible by 2. Odd Number: An integer n is called odd if it has the form n = 2 k +1 for some integer k. Prime Number: A natural number n > 1 is said to be prime if its only positive divisors are 1 and n. If n has other positive divisors, then n is called composite. Rational Number: A real number x is called rational if x can be written as a fraction m / n, where m and n are integers with n 0. Otherwise, x is called irrational. Direct Proof Consider the statements S 1! If p, then q, and S 2! " x, p( x). At times we may try to prove that these types of statements are true. To prove S 1 directly, we assume p is true and then argue that q must also be true. To prove S 2 is true, we pick an arbitrary x and argue that property p(x) must hold. Example 1. Prove the following results directly: (a) If n is an odd integer, then n 2 +1 is even. (b) If a divides b and b divides c, then a divides c. (c) For all rational numbers x and y, the product x y is also a rational number. Proof. (a) Assume n is an odd integer. Then n = 2 k +1 for some integer k. We then have n 2 +1 = (2k +1) = 4k 2 + 4k + 2 = 2(2k 2 + 2k + 1) = 2l, where l = 2k 2 + 2k + 1 is an integer because k is an integer. Thus, n 2 +1 has the form of an even number, and so n 2 +1 is even.

2 (b) Assume a divides b and b divides c. Then b = k a for some integer k, and c = j b for some other integer j. Thus, c = j b = j (ka) = ( j k )a. Because the product of integers j k is still an integer, we have that a divides c. (c) Let x and y be rational numbers. Then x = m / n and y = j / k for some integers m, n, j, and k, with n! 0 and k! 0. Then x y = m n! j k = m j n k. The products of integers m j and n k are still integers, and n k cannot be 0 because neither n nor k is 0. Thus, x y is in the form of a rational number and therefore x y is rational. Indirect Proofs Given an implication S! p " q, its contrapositive is ~ q! ~ p, which is logically equivalent to the original implication. In order to prove that S is true, it might be easier to prove that the contrapositive is true. That is, we can assume that q is not true, and then argue that p is not true. Another common method used to prove p! q is a proof by contradiction. In this case, we assume that p is true but that q is not true. We then argue that a mathematical contradiction occurs. We conclude that q in fact must be true. Proof by Contrapositive To prove p! q, assume that q is not true, then argue that p is not true. Proof by Contradiction To prove p! q, assume that p is true but q is not true. Then argue that a mathematical contradiction occurs. Notes: (i) A logical statement is often a conjunction a! b (a and b ) or a disjunction a! b (a or b ). We apply DeMorgan s Laws to obtain the negations of these statements as follows: ~ (a! b) " ~ a # ~ b ~ (a! b) " ~ a # ~ b (ii) The negation of one or the other but not both is given by both or neither. (iii) A logical statement may be in terms of a quantifier such as for every x, property p(x ) holds. The negation is there exists an x for which p(x ) does not hold. Statement:!x, p(x ) Negation:! x, ~ p(x )

3 Example 2. Prove the following results: (a) If c is a non-zero rational number, then c! is irrational. (b) If m 3 is even, then m is even. (c) For all real numbers x and y, if x + y > 0, then x > 0 or y > 0. (d) Let n be an integer. If n 2 is divisible by 3, then n is divisible by 3. (e) Let x be a real number. If x <! for all! > 0, then x = 0. (a) Proof. (By contradiction) Let c be a non-zero rational number, and suppose that c! is rational. Then c! = m / n, where m, n are integers and n 0. Moreover, c = j / k, where j, k are integers and k 0, but also j 0 because c is non-zero. We then have! = 1 c " m n = k j " m n = k m j n. The products of integers k m and j n are still integers, and j n cannot be 0 because neither n nor j is 0. Thus,! is in the form of a rational number, which is a contradiction because! is irrational. Ergo, c! must be irrational. (b) Claim: If m 3 is even, then m is even. We shall prove the contrapositive instead; we shall assume that m is not even and show that m 3 is not even. That is, we shall assume that m is odd and show that m 3 is odd. Proof. Assume that m is odd. Then m = 2 k +1 for some integer k. Then, m 3 = (2 k + 1) 3 = 8k 3 +12k 2 + 6k + 1 = 2(4k 3 + 6k 2 + 3k) + 1 = 2l + 1, where l is the integer 4k 3 + 6k 2 + 3k. Thus, m 3 is odd because it is written in the form of an odd integer. By contrapositive, if m 3 is even, then m is even. (c) Proof. Let x and y be real numbers. Assume that x! 0 and y! 0. Then by adding, we have x + y! = 0. That is, x + y! 0. Hence, x! 0 and y! 0 implies that x + y! 0. By contrapositive, if x + y > 0, then x > 0 or y > 0.

4 (d) Claim: For an integer n, if n 2 is divisible by 3, then n is divisible by 3. Proof. Assume n is not divisible by 3. We then can apply the Fundamental Theorem of Arithmetic to write n uniquely as a product of powers of primes as follows: n = p 1 r 1! p 2 r 2!...! p k r k, where r i are natural numbers and the p i are distinct primes. Because n is not divisible by 3, then p i 3 for 1! i! k. By squaring we obtain n 2 r = p 1 r 1! p 2 r ( 2!...! p k k ) 2 2r = p 1 2r 1! p 2 2r 2!...! p k k, which is a prime factorization of n 2. By uniqueness of prime factorization, the above p i for 1! i! k are the only prime factors of n 2. Because none of the p i equals 3, then n 2 is not divisible by 3. By contrapositive, if n 2 is divisible by 3, then n must also be divisible by 3. Note: The preceding result can be generalized as follows: Let p be prime and suppose k! 2. If n k is divisible by p, then n is divisible by p. (In the proof of (d), simply replace 3 with p and replace the exponent 2 with k.) (e) Claim: If x <! for all! > 0, then x = 0. Proof. Suppose x <! for all! > 0. We must show that x = 0. So suppose that x! 0. Then x < 0 or x > 0. In either case, we have x > 0. Now let! = x / 2 > 0. For this!, we then have x > x / 2 =!, which contradicts the assumption. Thus we must have that x = 0. Note: We also could have argued by contrapositive: Assume x! 0. Then we showed that there exists an! > 0 for which x <! fails. So if x! 0, then it is not true that x <! for all! > 0. By contrapositve, if it is true that x <! for all! > 0, then x = 0. Double Implication A double implication is of the form p! q, which is read p if and only if q. This statement means p! q and q! p and both implications must be proven. Example 3. Let n be an integer. Then n is even if and only if n 2 is even.

5 Proof. Suppose first that n is even. Then n = 2k, for some integer k. So then n 2 = (2k ) 2 = 4k 2 = 2(2k 2 ). Because 2k 2 is also an integer, we see that n 2 must be even. Dr. Neal, WKU Next we must prove that if n 2 is even, then n is even. But we shall prove the contrapositive instead. So assume that n is not even, i.e., that n is odd. Then n = 2k +1 for some integer k. We then have n 2 = (2k +1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) +1 = 2 j +1, where j = 2k 2 + 2k is an integer. Hence, n 2 is odd; that is, n 2 is not even. We have proven that if n is not even, then n 2 is not even. By contrapositive, if n 2 is even, then n is even. From both directions, we now have that n is even if and only if n 2 is even. Note: The second direction also follows from the generalization of Part (d) using the prime p = 2. In other words, if 2 divides n 2 then 2 divides n. Theorem. 2 is irrational. Proof. Assume 2 is rational. Then we can write 2 = m / n, where m and n are integers with n 0. Furthermore, we can assume that the fraction is reduced so that m and n have no common divisors. Then by squaring the fraction, we have 2 = m2 n 2 or 2n2 = m 2. Thus, m 2 is even because it is divisible by 2. It follows that the integer m must be even (by Example 3). So we may write m = 2k for some integer k. Then 2n 2 = m 2 = (2k) 2 = 4k 2 = 2(2k 2 ), which gives n 2 = 2k 2. Thus, n 2 is even because it is divisible by 2. It follows that the integer n also must be even. So both m and n are even and therefore both are divisible by 2. But this fact contradicts the assumption that we have chosen m and n to have no common divisors. This contradiction leads us to conclude that 2 cannot be written as a fraction. Thus, 2 is irrational. Corollary. For any prime p, p is irrational.

6 Disproving by Counterexample Consider the statements S = p! q and P =!x, p(x). At times we may try to prove that these types of statements are false. To do so, we can show that their negations are true, where ~ S = p! ~ q and ~ P =! x, ~ p(x ). To show that ~ S is true, we must demonstrate a case where p is true but q is false. To show that ~ P is true, we must exhibit an x for which p(x) fails. In other words, we must come up with counterexamples that disprove the original statements S and P. Example 4. Disprove the following results by finding counterexamples: (a) If f (x) is a continuous function, then f is one-to-one. (b) For all natural numbers n, if n is prime then 2 n!1 is prime. (c) For all real numbers x, 3 x > x 3. (d) For all integers m and n, if m n > 0, then m > 0 or n > 0 but not both. Solutions. (a) Let f (x) = x 2. Then f is a continuous function, but f is not one-to-one (because for instance f (2) = f (!2)). Thus, the statement is disproved. (b) The result holds for the prime numbers n = 2, 3, 5, and 7. But for n = 11, we have 2 11!1 = 2047 = 23 "89, which means that 2 11!1 is composite. Thus, there exists a natural number n = 11 such that 11 is prime but 2 11!1 is not prime. (c) Consider x = 3. Then 3 x = 3 3 = x 3. So there exists a real number x for which 3 x is not greater than x 3. Hence, the statement is disproven. (d) The implication is of the form S = p! (q " r). So its negation is ~ S = p! ~ (q " r) = p! ((q! r) " (~ q! ~ r) ) So to disprove the statement, we must show that there exixt integers m and n such that p is True together with both q and r, or p is True with neither q nor r. Now consider m = 2 and n = 3 which are both integers. Then m n > 0 ( p holds), and m > 0 and n > 0 (both q and r hold). Thus, we have a counterexample that disproves S. (We also could use an example such as m =!2 and n =!3.) Note: To disprove p! (q " r), we must find a case for which p! ~ (q " r) = p! (~ q! ~ r) is True. That is, p is True, but not q and not r. To disprove p! (q " r), we must find a case for which p! ~ (q! r) = p! (~ q" ~ r) is True. That is, p is True, but either not q or not r.

7 Exercises 1. Prove the following results in a formal, elegantly written, mathematical style: (a) For all integers m and n, if m is even and n is odd, then m + n is odd. (b) Let a be an integer. If a divides b, then b is an integer and a divides b 2. (c) If a divides b and a divides c, then a divides the linear combination mb + n c for all integers m and n. (d) If x and y are rational numbers, then x + y is a rational number. (e) Let m and n be integers. If m n is even, then m is even or n is even. (f) Let m and n be integers. If m + n is odd, then m is odd or n is odd but not both. (g) Let x and y be real numbers. Then x = y if and only if x! y < " for every! > 0. (h) For every prime number p, p is irrational. 2. Disprove the following claims: (a) For all real numbers x and y, if x y > 100, then x > 10 or y 11. (b) For all rational numbers x and y, if x y is an integer, then x is an integer and y is an integer. (c) If n is an odd integer, then n is even.

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs

The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### def: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.

Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true

### CS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers

CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x)

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### Math 3000 Section 003 Intro to Abstract Math Homework 2

Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### GREATEST COMMON DIVISOR

DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### 3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.

SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P

### 5-4 Prime and Composite Numbers

5-4 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee

### Pythagorean Triples Pythagorean triple similar primitive

Pythagorean Triples One of the most far-reaching problems to appear in Diophantus Arithmetica was his Problem II-8: To divide a given square into two squares. Namely, find integers x, y, z, so that x 2

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### 1.4 Factors and Prime Factorization

1.4 Factors and Prime Factorization Recall from Section 1.2 that the word factor refers to a number which divides into another number. For example, 3 and 6 are factors of 18 since 3 6 = 18. Note also that

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

### 12 Greatest Common Divisors. The Euclidean Algorithm

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### 3. Logical Reasoning in Mathematics

3. Logical Reasoning in Mathematics Many state standards emphasize the importance of reasoning. We agree disciplined mathematical reasoning is crucial to understanding and to properly using mathematics.

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Introduction to Proofs

Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are

### Proof of Infinite Number of Fibonacci Primes. Stephen Marshall. 22 May Abstract

Proof of Infinite Number of Fibonacci Primes Stephen Marshall 22 May 2014 Abstract This paper presents a complete and exhaustive proof of that an infinite number of Fibonacci Primes exist. The approach

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic 1 Introduction: Why this theorem? Why this proof? One of the purposes of this course 1 is to train you in the methods mathematicians use to prove mathematical statements,

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

### WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology

First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Mid-Session Test Summer Session 008-00

### Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### Elementary Algebra. Section 0.4 Factors

Section 0.4 Contents: Definitions: Multiplication Primes and Composites Rules of Composite Prime Factorization Answers Focus Exercises THE MULTIPLICATION TABLE x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5

### If n is odd, then 3n + 7 is even.

Proof: Proof: We suppose... that 3n + 7 is even. that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. that 3n + 7 is even. Since n is odd, there exists an integer k so that

### DigitalCommons@University of Nebraska - Lincoln

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University

### Today s Topics. Primes & Greatest Common Divisors

Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### 15 Prime and Composite Numbers

15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### Congruent Numbers, the Rank of Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture. Brad Groff

Congruent Numbers, the Rank of Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture Brad Groff Contents 1 Congruent Numbers... 1.1 Basic Facts............................... and Elliptic Curves.1

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### CHAPTER 1. Logic, Proofs Propositions

CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: Paris is in France (true), London

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### Day 2: Logic and Proof

Day 2: Logic and Proof George E. Hrabovsky MAST Introduction This is the second installment of the series. Here I intend to present the ideas and methods of proof. Logic and proof To begin with, I will

### COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

### Introduction. Appendix D Mathematical Induction D1

Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### Lesson 6: Proofs of Laws of Exponents

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 8 Student Outcomes Students extend the previous laws of exponents to include all integer exponents. Students base symbolic proofs on concrete examples to

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Reasoning and Proof Review Questions

www.ck12.org 1 Reasoning and Proof Review Questions Inductive Reasoning from Patterns 1. What is the next term in the pattern: 1, 4, 9, 16, 25, 36, 49...? (a) 81 (b) 64 (c) 121 (d) 56 2. What is the next

### Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the

### Propositional Logic. Definition: A proposition or statement is a sentence which is either true or false.

Propositional Logic Definition: A proposition or statement is a sentence which is either true or false. Definition:If a proposition is true, then we say its truth value is true, and if a proposition is

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### Theorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive

Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime

### Integers and division

CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Number of Divisors. Terms. Factors, prime factorization, exponents, Materials. Transparencies Activity Sheets Calculators

of Divisors Purpose: Participants will investigate the relationship between the prime-factored form of a number and its total number of factors. Overview: In small groups, participants will generate the

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### 5. Factoring by the QF method

5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

### IB Math Research Problem

Vincent Chu Block F IB Math Research Problem The product of all factors of 2000 can be found using several methods. One of the methods I employed in the beginning is a primitive one I wrote a computer

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### A Guide to Proof-Writing

by Ron Morash, University of Michigan Dearborn PW-1 At the end of Section 1.7, the text states, We have not given a procedure that can be used for proving theorems in mathematics. It is a deep theorem

### Prime Factorization 0.1. Overcoming Math Anxiety

0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF